The game uses twelve fair switches to form directed nonloop edges on vertices 0..3. The reward counts directed Hamiltonian cycles starting at 0, visiting each other vertex exactly once, and returning. Every policy using only revealed information has a 97% expected gross return. Simulation measurements are supporting observations, not the mathematical payout law.
Game Mechanics and Scoring
The core mechanic involves twelve independent fair switches that represent all directed nonloop edges on vertices 0..3. These switches are revealed in row-major edge or matrix order. The reward is the count of directed Hamiltonian cycles starting at vertex 0, visiting each of the other vertices exactly once before returning to the start.
The live gross quote is calculated as 0.97 times the conditional expected terminal reward, divided by its initial expected reward. This quote may rise or fall as switches are revealed. The 3% opening deduction appears only once in the quote.
- Zero Quote: Ends the round immediately.
- Positive Final Quote: Settles automatically.
Controls, Modes and Auto-Collect
Players can select one active stake from 10, 20, 50, or 100 virtual coins. Settings and refill locks apply during play.
Collection Rules
- Manual Collection: Available only after the first reveal.
- Auto Thresholds: Options for 1.2, 2, 3, or 5. These collect the actual full first-crossing quote. If a threshold is unreachable, the system waits for the terminal result.
A zero quote ends the round immediately, while a positive final quote settles at the finite trace cap.
Payout and Settlement Math
The actual payout math and every mode are recorded in the source-bound mathematical evidence. Displayed multipliers represent gross coin returns, not profit. The expected gross return is 97% for every policy using only revealed information.
| Audit collection policy | Expected gross return | Measured gross return |
|---|---|---|
| Collect after reveal 1 | 97.00% | 97.03% |
| Collect after reveal 6 | 97.00% | 96.96% |
| Collect after reveal 12 | 97.00% | 96.93% |
| First crossing 1.2× | 97.00% | 96.97% |
| First crossing 2× | 97.00% | 97.13% |
| First crossing 3× | 97.00% | 97.10% |
| First crossing 5× | 97.00% | 96.98% |
Each measurement uses 1,000,000 simulated rounds. Reveal-depth policies are audit examples of manual collection, not selectable stop modes. The automatic options are off, 1.2×, 2×, 3× and 5×. Results are rounded for readability; measured returns are sample observations, not session promises.
Fractional carry persists through settlement, manual refill, and reload where the actual contract specifies.
Wallet and Persistence
The demo wallet begins with an initial balance of 1000 virtual coins. Coins do not have cash value.
Restoration
If the idle balance falls below 10, it may be manually restored to exactly 1000. This is a manual set, not an automatic replenishment. Fractional change is preserved during this restoration.
Session Handling
- Settled Balance: Local settled balance including zero persists.
- Unfinished Round: If you reload the page, an unfinished round ends without refund or resume. It is not saved for later.
- Busy State: Stake, mode, and refill are locked during active play. Sound remains available.
Privacy and Storage
This guide describes only explicitly supplied source-observable storage. The game source uses localStorage for settled balance and fractional carry. This does not establish anonymity or exclude other processing.
The template declares a Google Fonts stylesheet; actual full-site network capture remains pending. The absence of account or payment controls in the demo does not prove that no personal data are processed.
Responsible Use and Audience
18+ is this guide's own intended audience policy, not a universal legal age or implemented age gate.
Responsible-use pages can describe optional pauses, a personal time or virtual-coin limit, and finding appropriate local qualified help. The demo does not clinically train behavior or make losses safe. There is no promise of account-level blocking, a timer, or self-exclusion tools within this specific demo interface.
Frequently asked questions
What is the expected return for Directed Hamilton Gate?
Every policy using only revealed information has a 97% expected gross return. This is a mathematical expectation, not a guaranteed session result. Measured RTP in simulations may vary slightly (e.g., 97.025% vs 97.000% in mode stop=1).
How does the auto-collect feature work?
Auto thresholds are set to 1.2, 2, 3, or 5. They collect the actual full first-crossing quote. If the threshold is unreachable before the end of the trace, the system waits for the terminal result. A zero quote ends the round immediately.
What happens if I reload the page during a game?
An unfinished round ends without refund or resume. The balance persists as settled, but the current unrevealed switches and progress are lost. You must start a new round.
